Olefin oxides represented by propylene oxide are industrially important as intermediate materials of industrial chemicals, synthetic resins, rubbers or the like. As the process for producing such an olefin, non-patent Document 1 (p. 95, FIG. 1) discloses a process in which olefin is reacted with oxygen in the presence of a catalyst containing silver supported on calcium carbonate and potassium as an promoter and that about 2% of a potassium promoter contained in the catalyst is important to improve selectivity of propylene oxide and decrease of potassium content in the catalyst led to remarkable decrease of the selectivity. Patent Document 1 discloses an oxidation of propylene using a catalyst containing a potassium promoter. In addition, a catalyst containing a silver salt supported on a silicate in which sodium silicate is used as a raw material, and a silver catalyst containing silver supported on alumina are known (Patent Documents 2 and 3). Furthermore, a process for producing an olefin oxide, in which water is added to a reaction gas is also disclosed (Patent Document 4). However, selectivity of olefin oxide in these processes is not always satisfactory, and a further improvement has been required from an industrial point of view.    Patent Document 1: JP-T 2002-510306    Patent Document 2: JP-A 1-231942    Patent Document 3: GB 1,368,922    Patent Document 4: JP-A 2004-307485    Non-patent Document 1: Catalysis Letters, June 2002, Vol. 80, 3-4, pp. 93-98